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Theorem riinint 5107
Description: Express a relative indexed intersection as an intersection. (Contributed by Stefan O'Rear, 22-Feb-2015.)
Assertion
Ref Expression
riinint  |-  ( ( X  e.  V  /\  A. k  e.  I  S 
C_  X )  -> 
( X  i^i  |^|_ k  e.  I  S
)  =  |^| ( { X }  u.  ran  ( k  e.  I  |->  S ) ) )
Distinct variable groups:    k, V    k, X
Allowed substitution hints:    S( k)    I(
k)

Proof of Theorem riinint
StepHypRef Expression
1 ssexg 4567 . . . . . . 7  |-  ( ( S  C_  X  /\  X  e.  V )  ->  S  e.  _V )
21expcom 436 . . . . . 6  |-  ( X  e.  V  ->  ( S  C_  X  ->  S  e.  _V ) )
32ralimdv 2835 . . . . 5  |-  ( X  e.  V  ->  ( A. k  e.  I  S  C_  X  ->  A. k  e.  I  S  e.  _V ) )
43imp 430 . . . 4  |-  ( ( X  e.  V  /\  A. k  e.  I  S 
C_  X )  ->  A. k  e.  I  S  e.  _V )
5 dfiin3g 5104 . . . 4  |-  ( A. k  e.  I  S  e.  _V  ->  |^|_ k  e.  I  S  =  |^| ran  ( k  e.  I  |->  S ) )
64, 5syl 17 . . 3  |-  ( ( X  e.  V  /\  A. k  e.  I  S 
C_  X )  ->  |^|_ k  e.  I  S  =  |^| ran  (
k  e.  I  |->  S ) )
76ineq2d 3664 . 2  |-  ( ( X  e.  V  /\  A. k  e.  I  S 
C_  X )  -> 
( X  i^i  |^|_ k  e.  I  S
)  =  ( X  i^i  |^| ran  ( k  e.  I  |->  S ) ) )
8 intun 4285 . . 3  |-  |^| ( { X }  u.  ran  ( k  e.  I  |->  S ) )  =  ( |^| { X }  i^i  |^| ran  ( k  e.  I  |->  S ) )
9 intsng 4288 . . . . 5  |-  ( X  e.  V  ->  |^| { X }  =  X )
109adantr 466 . . . 4  |-  ( ( X  e.  V  /\  A. k  e.  I  S 
C_  X )  ->  |^| { X }  =  X )
1110ineq1d 3663 . . 3  |-  ( ( X  e.  V  /\  A. k  e.  I  S 
C_  X )  -> 
( |^| { X }  i^i  |^| ran  ( k  e.  I  |->  S ) )  =  ( X  i^i  |^| ran  ( k  e.  I  |->  S ) ) )
128, 11syl5eq 2475 . 2  |-  ( ( X  e.  V  /\  A. k  e.  I  S 
C_  X )  ->  |^| ( { X }  u.  ran  ( k  e.  I  |->  S ) )  =  ( X  i^i  |^|
ran  ( k  e.  I  |->  S ) ) )
137, 12eqtr4d 2466 1  |-  ( ( X  e.  V  /\  A. k  e.  I  S 
C_  X )  -> 
( X  i^i  |^|_ k  e.  I  S
)  =  |^| ( { X }  u.  ran  ( k  e.  I  |->  S ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1868   A.wral 2775   _Vcvv 3081    u. cun 3434    i^i cin 3435    C_ wss 3436   {csn 3996   |^|cint 4252   |^|_ciin 4297    |-> cmpt 4479   ran crn 4851
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-sep 4543  ax-nul 4552  ax-pr 4657
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-ral 2780  df-rex 2781  df-rab 2784  df-v 3083  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3910  df-sn 3997  df-pr 3999  df-op 4003  df-int 4253  df-iin 4299  df-br 4421  df-opab 4480  df-mpt 4481  df-cnv 4858  df-dm 4860  df-rn 4861
This theorem is referenced by:  cmpfiiin  35458
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